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Bayesian Model Averaging for Estimating theTemperature Distribution in a Steam Methane

Reforming Furnace

Anh Tran1, Madeleine Pont1, Andres Aguirre1, Helen Durand1,Marquis Crose1 and Panagiotis D. Christofides1,2

1 Department of Chemical & Biomolecular Engineering, University of California, LosAngeles

2Department of Electrical Engineering, University of California, Los Angeles

Inaugural Data Science Workshop

August 7, 2018

Anh Tran Inaugural Data Science Workshop August 7, 2018 1 / 20

Uses and Applications of Hydrogen Gas

Hydrogen is one of the most important raw materials for thepetroleum refinery industry (Gupta, CRC Press, 2008)

Olefins +H2 (g) −→ ParaffinsR1−H2C−CH2−R2 (g)+H2 (g) −→ R1−H2CH (g)+HCH2−R2 (g)R − SH (g) + H2 (g) −→ R (g) + H2S (g)

Hydrogen is a precursor for many chemical industries, e.g.,ammonia production

3H2 (g) + N2 (g)∆H≪0−−−−→ NH3 (g)

Hydrogen is a carrier gas for the production of thin film solarcells (Crose et al., Chem. Eng. Science , 2015)

e− + H2 (g) −→ e− + 2H·

H· + SiH4 (g) −→ H2 (g) + (SiH3)·

Hydrogen is an efficient energy carrier for hydrogen-basedtechnologies (e.g. fuel cells)


General Information of Hydrogen Production

In industry, hydrogen is produced by

Steam methane reforming (SMR) process, which accounts for 48%of world-wide hydrogen production (Ewan and Allen, Int. J. Hydrogen Energy, 2005)

CH4(g) + H2O(g) ↼−−−−−−−−⇁∆H≫0NiAl2O3

CO(g) + CO2(g) + H2(g) (1)

Top-fired reformer


Industrial-scale Steam Methane Reformer

GeometryLength: 16 mWidth: 16 mHeight: 13 m

Components336 reforming tubes96 burners8 flue-gas tunnels

Daily hydrogen production of2.8×106 Nm3

Daily superheated steamproduction of 1.7×106 kg

Annual operating cost of $62×106


Industrial-scale Steam Methane Reformer Mesh

The industrial-scale reformer mesh consists of 41 million grids

Mesh information

The reformer meshRecommendedrange

Min orthogonal factor 0.459 0.167 − 1.000

Max ortho skew 0.541 0.000 − 0.850


Industrial-scale Reformer CFD Model

Modeling turbulent flowsStandard k − ǫ model with ANSYS enhanced wall treatment

Modeling the combustion phenomenaPremixed combustion assumptionGlobal kinetic model of CH4 combustion (D. G. Nicol, PhD Thesis, 1995)

Global kinetic model of H2 combustion (Bane et al., Technical Report, 2010)

FR/ED turbulence-chemistry interaction modelModeling thermal radiation

Empirical model for radiative properties (A. Maximov, PhD Thesis, 2012)

Beer’s law and Kirchoff’s lawDiscrete ordinate method


Industrial-scale Reformer CFD Model

Modeling turbulent flowsStandard k − ǫ model with ANSYS enhanced wall treatment

Modeling the catalyst network of each reforming tubeA continuum approach using ANSYS porous zone functionEffectiveness factor and catalyst packing factor

Modeling the tube wall of each reforming tubeANSYS thin wall function

Modeling the SMR processGlobal kinetic model (J. Xu and G. F. Froment, AIChE Journal, 1989)


CFD Model Validation with Plant Data

Simulation data generated by the reformer CFD model is in goodagreement with the data provided by industry

Reformer CFD model Industry Deviation

Fired duty (kW) 209474.8 211597.3 1.0 %

Total absorbedheat (kW)

113895.5 112246.2 1.5%

Fraction ofabsorbed heat (%)

54.4 53.1 2.4%

OD averageheat flux (kW/m2)

59.2 58 2.1%

ID averageheat flux (kW/m2)

69.5 75.7 8.2%

Average outletflue gas temp (K)

1243.1 1283 3.1%

x̄outletH2

46.5 46.8 0.6 %


Motivations for Data-driven Modeling

The reformer service life is monitored by the system of infraredcameras that periodically record the outer wall temperatures(OTWTs) of the reforming tubes in real-time

Feedback from our third-party collaborator and publicly availableliterature suggest that OTWTs can be controlled by the total fuelflow rate and its spatial distribution inside the reformer

We developed an integrated model identification procedure todiscover the dependence of the OTWT distribution on the reformerinput using

Bayesian variable selection

Sparse nonlinear regressionBayesian model averaging

Theories of thermal radiation

Reformer geometry


Data-driven Model for the ith OTWT

The data-driven model for therelationship between the i th

OTWT at a fixed height and thefurnace-side feed (FSF)distribution is formulated asfollows,

T̂ P,ni =

Ki∑

k=1

wPi,k T̃ P,n

i,k (2a)

where

Ki∑

k=1

wPi,k = 1 (2b)

T̃ P,ni,k =

G∑

g=1

(#»α

kgi

)T· fg

(#»

F n)+ αk

i (2c)

#»

F n =[F n

1 ,Fn2 , · · · ,F

n96

]T (2d)∥∥∥

#»

F n∥∥∥

1= F n

tot (2e)

T̂ P,ni is the BMA estimate of the ith

OTWT

T̃ P,ni,k is the estimate of the ith OTWT

based on the kth sub-model for theith OTWT

fg(

#»

F n)

is the gth basis function in

the library of transformation functions


k th Sub-model for the ith OTWT (i.e., Mi,k )

T̃ P,ni ,k =

G∑

g=1

(#»α

kgi

)T· fg

(#»

F n)+ αk

i (3)

where

f1(

#»

F n)=

[F n

1 ,Fn2 , · · · ,F

n96

]T

f2(

#»

F n)=

[(F n

1

)2,(F n

2

)2, · · · ,

(F n

96

)2]T

f3(

#»

F n)=

[(F n

1

)3,(F n

2

)3, · · · ,

(F n

96

)3]T

f4(

#»

F n)=

[2√

F n1 ,

2√

F n2 , · · · ,

2√

F n96

]T

f5(

#»

F n)=

[3√

F n1 ,

3√

F n2 , · · · ,

3√

F n96

]T

f6(

#»

F n)=

[4√

F n1 ,

4√

F n2 , · · · ,

4√

F n96

]T

f7(

#»

F n)=

[5√

F n1 ,

5√

F n2 , · · · ,

5√

F n96

]T

f8(

#»

F n)=

[exp

(F n

1

), exp

(F n

2

), · · · , exp

(F n

96

)]T

G = 8 is the number of functions in the library of transformation functions

#»αkgi ∈ IR96×1 is the empirical vector of Mi,k corresponding to fg (·)

αki ∈ [298.15, 348.15] is the estimated ambient temperature of Mi,k


Spare Nonlinear Regression with MLE

The formulation for the sparse nonlinear regression with MLE isproposed as follows

minαk

i ∈[298.15,348.15]α

kgij ∈[0,∞)

N∑

n=1

(T n

i − T̃ P,ni,k

)2

2(σn

i

)2 + λi

8∑

g=1

∥∥∥ #»αkgi

∥∥∥1

(6)

subject to8∑

g=1

αkgil fg

(F

0)=

8∑

g=1

αkgij fg

(F

0)

if dil = dij (7a)

8∑

g=1

αkgil fg

(F

0)≥

(dij

dil

)βl 8∑

g=1

αkgij fg

(F

0)

if dil < dij (7b)

8∑

g=1

αkgil fg

(F

0)≤

(dij

dil

)βu 8∑

g=1

αkgij fg

(F

0)

if dil < dij (7c)

F0=

F typtot

96(7d)


Spare Nonlinear Regression with MLE

Regressors of the ith OTWT are defined asthe burners that directly control the ith OTWT

T̃ P,ni ,k =

G∑

g=1

(#̂»α

kgi

)T· fg

(#»

F n∣∣∣SiR

)+ α̂k

i (8)

SiR ∈ IRj×1 is the given set of regressors

#̂»αkgi is the MLE of #»α

kgi ∈ IRj×1

α̂ki is the MLE of αk

i ∈ IR

#»

F n∣∣∣SiR

∈ IRj×1 is the design matrix


Model Identification Flow Diagram


Simulation Inputs

21 reformer CFD data sets with varying FSF distributions and totalFSF flow rates are availableReformer CFD data sets are partitioned into two categories

A reformer CFD training data consists of 18 data setsA reformer CFD testing data consists of 3 data sets

Sλ = {0.1,0.2, · · · ,1.0,1.2, · · · ,2.0,5.0,10,20}, which controlsthe model complexity and goodness of fit

Small values of λi result in a low degree of shrinkage and favoroverfitting data-driven models with high goodness of fitLarge values of λi result in a high degree of shrinkage and favorunderfitting data-driven models with low levels of complexity

Leave-one-out cross validation is used to find the best λi

The 77th reforming tube is chosen as a representative examplebecause the number of sub-models with high goodness of fit (i.e.,4) and the number of predictors (i.e., 9) for the 77th OTWT


Results

0.1 1 10LASSO parameter (λ77)

2

3

4

Mea

n sq

uare

d er

ror

(K2 ) Prediction error

Fitting error

The smallest fitting error correspondsto the smallest value of λ=0.1

The largest fitting error correspondsto the largest value of λ=10

The smallest prediction errorcorresponds to λ̂77=0.5

0 5 10 15 20Index of reformer CFD simulation

1100

1150

1200

1250

77th O

TW

T (

K)

CFD dataTraining dataForecasting data

The model for the 77th OTWT

T̂ P,n77 =0.01T̃ P,n

77,1 + 0.23T̃ P,n77,2

+ 0.29T̃ P,n77,3 + 0.47T̃ P,n

77,4

(9)


Results

The data-driven model for the OTWT distribution correctly identifies thehot and cold regionsThe absolute maximum and average deviations from the reformer CFDdata are 20.1 K and 2.9 K, respectively


Flowchart of The Furnace-balancing Scheme

The furnace-balancing scheme isdeveloped based on

The high-fidelity reformer CFD model(Tran et al., Chem. Eng. Sci., 2017)

The statistical-based modelidentification (Tran et al., Chem. Eng. Res. Des., in press)

The furnace-balancing optimizer (Tran et al.,

Comp. & Chem. Eng., 2017)

0 2 4 6 8 10 12Distance down the reforiming tube (m)

800

900

1000

1100

1200

1300

Tem

pera

ture

(K

)

Desgin outer wall temperatureMaximum outer wall temperatureAverage outer wall temperatureMinimum outer wall temperature

0 2 4 6 8 10 12Distance down the reforiming tube (m)

800

900

1000

1100

1200

1300

Tem

pera

ture

(K

)

Desgin outer wall temperatureMaximum outer wall temperatureAverage outer wall temperatureMinimum outer wall temperature


Results

This result is obtained within a minute

The total FSF flow rate is increased by 40% from 98.113 to136.896 kg sec−1 without damaging the reforming tubes indicatedby the evidence that the maximum value in the OTWT distributionis 1288.35 K

0 1 2 3 4 5Iteration of the Heuristic Search

100

120

140

Tot

al F

uel F

low

Rat

e (k

g s-1

)

0 2 4 6 8 10 12Distance from the Ceiling (m)

900

1000

1100

1200

1300

OT

WT

(K

)T

maxwall

Tave

wall

Tmin

wall


Conclusion

The integrated model identification procedure is structured to befully distributed, which allows the data-driven model for 336reforming tubes to be derived simultaneously from the trainingdata and independently from one another

Leave-out-one cross validation is successfully implemented to findthe optimal LASSO parameter for each reforming tube

The results from the goodness-of-fit and out-of-sample predictiontests of the data-driven model for the OTWT distributiondemonstrated the high effectiveness of the method proposed inthis work

AcknowledgmentsFinancial support from the Department of Energy is gratefullyacknowledged


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